ends of a group
Walk off to infinity in a group's Cayley graph and ask: in how many essentially different directions can you escape? On a line you can go two ways, left or right — two ends. On a plane grid all the far-away regions are connected around the outside — there is just one end. The number of ends counts the connected ‘ways to infinity’ that remain after you delete any large but finite chunk and look at what is left.
Precisely, the number of ends of a finitely generated group G is the supremum, over all finite subsets K of its Cayley graph, of the number of infinite connected components of the complement of K — a quantity independent of the generating set and invariant under quasi-isometry. A remarkable theorem of Hopf (refined by Freudenthal) shows the answer is always 0, 1, 2, or infinity: no group has exactly three ends.
Each value has a structural meaning. Zero ends means the group is finite. Two ends means the group is virtually infinite cyclic (virtually Z). Infinitely many ends, by Stallings' celebrated splitting theorem, means the group splits nontrivially as an amalgamated free product or HNN extension over a finite subgroup — geometry forcing an algebraic decomposition. One end is the generic, ‘connected at infinity’ case, including Z^2, surface groups, and most groups one meets.
Z has two ends (left and right). Z^2 has one end (the grid is connected around any finite box). The free group F_2 has infinitely many ends — its Cayley tree breaks into more and more pieces as you delete larger balls.
Ends count 2 for Z, 1 for Z^2, infinitely many for a free group — never exactly 3.
Stallings' theorem characterizes groups with more than one end via splittings over finite subgroups; it underlies the proof that finitely generated groups of cohomological dimension 1 are free.